Evaluation functions and composition operators on Banach spaces of holomorphic functions

Abstract

Let $B(\Omega)$ be the Banach space of holomorphic functions on a bounded connected domain $\Omega$ in $\mathbb C^n$, which contains the ring of polynomials on $\Omega $. In this paper, we first establish a criterion for $B(\Omega )$ to be reflexive via evaluation functions on $B(\Omega )$, that is, $B(\Omega )$ is reflexive if and only if the evaluation functions span the dual spaces $(B(\Omega ))^{*} $. Moreover, under suitable assumptions on $\Omega$ and $B(\Omega)$, we establish a characterization of the composition operator $C_\varphi$ to be a Fredholm operator on $B(\Omega)$ via the property of the holomorphic self-map $\varphi:\Omega\to\Omega$. Our new approach utilizes the symbols of composition operators to construct a linearly independent function sequence, which bypasses the use of boundary behavior of reproducing kernels as those may not be applicable in our general setting.